Abstract

We consider the $d$-dimensional nonlinear Schrödinger equation under periodic boundary conditions: \[ -i\dot u=-\Delta u+V(x)*u+\varepsilon \frac{\partial F}{\partial \bar u}(x,u,\bar u), \quad u=u(t,x),\;x\in\mathbb{T}^d \] where $V(x)=\sum \hat{V}(a)e^{i\langle a,x\rangle}$ is an analytic function with $\hat V$ real, and $F$ is a real analytic function in $\Re u$, $\Im u$ and $x$. (This equation is a popular model for the ‘real’ NLS equation, where instead of the convolution term $V*u$ we have the potential term $Vu$.) For $\varepsilon=0$ the equation is linear and has time–quasi-periodic solutions \[ u(t,x)=\sum_{a\in \mathcal{A}}\hat u(a)e^{i(|a|^2+\hat{V}(a))t}e^{i\langle a,x\rangle}, \quad |\hat u(a)|>0, \] where $\mathcal{A}$ is any finite subset of $\mathbb{Z}^d$. We shall treat $\omega_a=|a|^2+\hat V(a)$, $a\in\mathcal{A}$, as free parameters in some domain $U\subset\mathbb{R}^{\mathcal{A}}$.

This is a Hamiltonian system in infinite degrees of freedom, degenerate but with external parameters, and we shall describe a KAM-theory which, under general conditions, will have the following consequence:

If $|\varepsilon|$ is sufficiently small, then there is a large subset $U’$ of $U$ such that for all $\omega\in U’$ the solution $u$ persists as a time–quasi-periodic solution which has all Lyapounov exponents equal to zero and whose linearized equation is reducible to constant coefficients.